Fuzzy Sets and Systems 61 (1994) 137-142
North-Holland
137
New operations defined over the
intuitionistic fuzzy sets
Krassimir T. Atanassov
Math. Research Lab., IPACT. Sofia, Bulgaria
Received November 1992
Revised April 1993
Abstract." New types of operations are defined over the intuitionistic fuzzy sets.
Keywords: Fuzzy set; intuitionistic fuzzy set.
Some operations (as U, N, +, .) are defined over the Intuitionistic Fuzzy Sets (IFSs) in [l]. Here we
shall introduce four new ones, and we shall show some of their basic properties.
Let a set E be fixed. An IFS A in E is an object having the form
A* = {(x, ]~A(X), ~IA(X)) I X E E},
where the functions I~A:E ~ [0, 1] and "YA: E ~ [0, 1] define the degree of membership and the degree
of non-membership of the element x • E to the set A, which is a subset of E, respectively, and for every
x•E:
0 ~ I..£A(X) Jr ~/z(X) ~ 1.
For every two IFSs A and B are valid (see [1-5]) the following definitions (let a , / 3 • [0, 1]):
Z c B
iff (Vx • E) (I~A(X) <~tZB(X) & yA(X) >~ yB(X)),
A~B
iffBcA,
A = B
iff (Vx e E) (tXz(X) = ~B(X) & yA(X) = yB(X)),
~-- {<X, ~IA(X), ],ZA(X))IX ~- E},
A n B = {<x, min(/.ta(X), /xB(x)), max(ym(X), yB(x))) I x • E},
A U B = {(x, max(~a(x), tZB(X)), min(ym(X), yB(x))) I x • E},
A + B = {<x, I~A(X) + tZB(X) -- tXz(X)" tZs(X), yA(X)" yB(X)> I X • E},
A- B -~-{<x, I~A(X) • I,~B(X), ~IA(X) -[- "YB(X) -- ~/A(X) " ~IB(X)) IX E E},
[qA = {<x,/zA(x), 1 - / x A ( x ) ) I x E E},
~ A = {<x, 1 - yA(x), yA(X)>]X E E},
C(A) = {<x, K, L> I x • E}
where K = max ~A(X),
xeE
L = min yA(X),
xEE
Correspondence to: Dr. K.T. Atanassov, Math. Research Lab., IPACT, P.O. Box 12, Sofia, Bulgaria.
0165-0114/94/$07.00 Q 1994---Elsevier Science B.V. All rights reserved
SSDI: 0165-0114(93)E0151-H
K. I". Atanassov / lntuitionistic f u z z y sets
138
I ( A ) = {(x, k, l) Ix e E}
where k = min
x~E
[,.tA(X),
l = max yA(X),
xEE
D~(A ) -- {(x, ~ A(X ) + a . Z A(X ), y A(X ) + (1 -- a ) " Z A(X )) ] X • E},
F~,t3(A ) : {(x, tZA(X) + a " teA(X), yA(X) + fl " ZA(X)) ] X • E}
G~,~(A) = {<x, ~ - ~ A ( X ) ,
where a +fl ~< 1,
tS" ~A(X)>IX • E},
H,~,~(A) = {(x, ~ " ~ A ( x ) , ~o(x) + t3" ~A(X)) [ x ~ e},
H*,tffA) = {(x, ~ " ~A(X), y~(X) + /S" (1 -- ~ " ~ ( X )
-- ~A(X))) IX • E},
J,~,t3(A) = {(x, tZA(X) + C~ • teA(X), ~ " yA(X)) I x • E},
J*,t~(A) = {(x, I.tA(S) + a • (1 - tZA(X) -- fl • yA(X)), fl " yA(X)) I x • E},
X a , b , c , d , e J ( A ) = {(X, a " tZA(X)
+ b" (1 - IZA(X) -- C" yA(X)),
d" 7A(X) + e " ( l - f w h e r e a , b , c , d , e , f e [ O , 1]
~A(X) -- yA(X))) I X C g},
anda+e-e.f<~l,
b+d-b.c<~l.
Here we shall define the following new operations (cf. [6-10]):
A @ B : {x, ~(tZA(X) + IZB(X)), I(yA(X) + ys(X)) I x • E},
A $ B = {(x, X/I.tA(X ) • tZB(X), X/yA(X)" ys(X)) [ X • E},
A # S = {(x, 2" t-tA(X)" tXB(X)/(IZA(X) + IZB(X)), 2" yA(X)" yB(X)/(yA(X) + yB(X))) ] X • E}
for which we shall accept that if /zA(x)=/z.(x) = 0, then /ZA(X)" IZB(X)/(IZA(X)+ /ZB(X))= 0 and if
yA(X) = yB(X) = 0, then yA(X) " yB(X)/(yA(X) + ")'8(X)) = 0;
A*B
t \ x ' 2 • (~-~-~ : ~ +
1)'2" (yA(X): yB---~ + 1)-/ x • .
Obviously, for every two IFSs A and B, A @ B, A $ B, A # B and A * B are an IFS.
Theorem 1. For every three 1FSs A, B and C:
(a) A @ B = B @ A ,
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(1)
(m)
(n)
(o)
(p)
A$B=B$A,
A#B=B#A,
A*B=B*A,
,4@/~ = A @B,
ASB=A$B,
~#B:A#B,
A*B-=A*B,
(AAB)@C=(A@C)N(B@C),
(AUB)@C=(A@C)U(B@C),
(ANB)#C=(A#C)A(B#C),
(AUB)#C--(A#C)U(B#C),
(A+B)@C~(A@C)+(B@C),
(A.B)@C=(A@C).(B@C),
(A@B)+C=(A+C)@(B+C),
(A @ B ) . C = (A. C ) @ ( B . C).
K.T. Atanassov / Intuitionisticfuzzy sets
139
PrOOf. (m) Initially we shall prove, that for every three numbers a, b, c ~ [0, 1] the following
inequality is valid:
c .(2-a-b
-c)+a
.b>~O.
(*)
When c 2 ~<a • b, then
c" ( 2 - a - b - c ) + a
•b =c. (2-a -b)-c2
+a .b
~ a .b-c2>~O.
When c z > a • b, then
c.(2-a-b-c)+a.b
>~Va . b . ( 1 - a - b + V~a . b)
>_ ~'V-d-a-b. ( ( 1 - a ) + V b . (Vaa- Vb))~> 0 ifa>~b,
~ [V~-a• b • ((1 - b) + V-da• (X/-b- V~a)) ~>0
if a < b .
Let A, B and C be three given IFSs. Then
(A + B) @ C = {(x, ½(tXA(X) + tXB(X) -- tZA(X) " tZB(X) + IZc(X)), ½(3'A(X) " 3JB(X) + 3~c(X))) I X E E}
(A @ C ) + (B @ C ) =
{(X, I([tZA(X ) -~- I,£c(X)) + I([,ZB(X ) + IJbc(X)) -- I(I&A(X ) + ]£c(X)) " ([£B(X) "j- ~ c ( X ) ) ,
I(')IA(X ) + ~l(,(X))" (~fB(X) -~ ")/c(X))} IX ~_ E}.
Using (*), from
½([£A(X) + [,£c(X)) -~- I([.£B(X) -~- [,.Lc(X)) -- I(~LA(X) + ~Lc(X)) " (~.LB(X) + ~Lc(X))
½(
A(x) +
-
+
= 1(2" [£c(X) + I,£A(X)" I.LB(X) -- [£A(X)" I£c(X) -- ]£B(X)" ],£c(X) -- /,£c(X) 2) ~ 0
and from
l(~/A(X ) " ~/B(X) + "Yc(X)) -- I(~/A(X ) + "yc(X))" ('yB(X) + ~Ic(X))
= 1(2" ~/c(X) + ~tA(X ) " ~/B(X) -- ~/A(X)" ~/c(X) -- ~/B(X)" ")Ic(X ) -- "yc(X) 2) ~ 0.
we see that (m) is valid.
The other assertions are proved in a similar way. The other relations between the operations @, $, # ,
• and the other above operations are not valid. By the same means the following theorem is proved.
Theorem 2. For every two IFSs A and B and for every two numbers a, ~ E [0, 1]:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
[](A @ B) = NA @ NB,
[S](A $ B) ~ S A $ [3B,
N(A # B) ~ [BA # [3B,
O(A @ B) = OA @ OB,
O(A$B)cOA$<>B,
O(A # B) ~ OA # OB,
D~(A @ B) = D~(A) @ D~(B),
F~,~(A@B)=F,,.t~(A)@F~,~(B ),
(i) G~.~(A @ B) = G,.,a(A) @ G.,~(B),
(j) G,,~(A $ B) = G,,~(A ) $ G,,~(B ),
(k) G,,~(A # B) = G,,~(A) # G,,~(B),
(l) H..~(A @ B) = H,~.z(A) @ H,~.~(B),
for a +/3~<1,
140
K.T. Atanassov / lntuitionisticfuzzy sets
(m) H*t~(A @ B) = H*.t~(A ) @ H*,t~(B ),
(n) J.,t3(A @ B) = J,~,t~(A)@J~,t~(B),
(p) J*~,~(A @ B ) = J*~,z(A) @J~* ~(B),
(q) Xa, b,c,d,e,f(A @ B) = Xa, b,c,d,e,/(A) @Xa,b,c,d,e,f(B),
(r) C(A @ B) c C(A) @ C(B),
(s) C(A $ B ) = C(A) $ C(B),
(t) I(A @ B) = I(A) @ I(B),
(u) I(A $ B ) = I ( A ) $ I ( B ) .
It can be easily seen that the equalities
(A@B)@C=A@(B@C),
(ASB)$C=A$(B$C),
(A#B)#C=A#(B#C),
(A*B)*C=A*(B*C)
are not valid. Thus we define
2
A1 @ A2 = @
i=1
Ai
and
i=1
Ai = {<x,(/~l ~'LAi(X))/n, (~1 "YAi(X))/n> x (~:E}.
.=
.=
Now, we have the following theorem.
Theorem 3.
For every n + 1 IFSs A1, A2, • • •, An and B:
(a)@Ai=@Ai,
i=l
i=l
(b)@ Ai
+ B = @ (Ai + B),
i=1
i=1
(c) @ Ai
" B = + (ai " B).
i=1
Theorem 4.
(a) []
(d)F~,fl
i=1
For every n IFSs ml, A2, . . . , A , and for every two numbers a, ~ e [0, 1]:
Ai
=
•Ai,
i=l
F~,,t3(Ai), for a + fl ~ l,
Ai =
i=l
K.T. Atanassov / Intuitionisticfuzzy sets
(f) H,,t3
A, =
141
H,,t~(A, ),
i=1
(g)H*~
As =
H*,¢(Ai ,
•=
(h)Jo.~
i=l
A, =
J~.~ (A,),
i=l
(i)
*
J,,,t~
Ai
*
J~,t3(Ai),
=
i=1
The proofs of these assertions are the same as in the above proof.
Theorem
5. For every two IFSs A and B:
{
(a)A. B c
[A#B}
ANBciA$B
I A@B
A.B
}
cA+B,
cAUB
(b)m $ B c D A @B,
(c)A@BcoA$B,
(d)A # B c D A $ B c D A
@ B,
(e)A @ B c o A $ B c o A
# B,
(f) A * B c E A @B,
(g)A * B c o A # B.
The validity of these inclusions follows from the validity of the following inequalities for arbitrary
real numbers a, b E [0, 1]:
a.b<~min(a,b)<~
a .b<~
a+b
2.(a-b+l)
2"a.b
a+b
<~ aVaT-~.b<~--<~max(a,b)<~a+b-a.b,
a+b
2
a+b
<~--
2
These operations can be transfered to operations on ordinary fuzzy sets as follows:
A @ B = {(x, l(]dbA(X) -~- [.LB(X)) ) IX E E},
A $ B = {(x, VbtA(X) • ~e(X)) I X E E},
A # B = {(x, 2- IXA(X)" t-tB(X)/(tZA(X) + ~,(X))) I x E E},
A.B=
X, 2.(/~A(X).tXs(X)+I)
x•E
.
K.T. Atanassov / lntuitionistic fuzzy sets
142
NOW, T h e o r e m 5 o b t a i n s t h e f o r m :
Z .B~
{ANBcA*BcA$BcA@BcAUB}A.B
cA+B.
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